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Fractal of the Day
by Jim Muth
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Fractal enthusiasts and visionaries: I named today's fractal picture "Art Nouveau Midget" because it reminds me of the curvilinear art style that was popular around the turn of the last century. It is the third in the present series of spectacular midgets. How long can I keep the streak going? I haven't the slightest idea, since the quality of the FOTD images is largely a matter of good luck in searching. The formula behind the image is Z^(-1.05)-Z^(1.05)+C, an expression that I entered on a whim and examined with more intuition than logic. The parameter file runs in under four minutes -- still slow enough to make the download preferable. That download can be found at: The fractal weather today was cloudy with off and on flurries of snow, amounting to about 1cm. The temperature of 36F kept the cats indoors. Our friend Percy Smedley is still puzzling over that four-dimensional figure painted into the surface volume of his west wall. The object, if that word can be applied to a 4-D thing, is the fourth of the six regular 4-D polytopes. The monstrous thing consists of 24 octahedral cells, 96 triangular faces, 96 edges and 24 vertices. The three dimensional projection, which resembles an octahedron filled with a network of distorted octahedra, is too complex to describe in detail. To form the figure, a hypercube must be inscribed in a hypersphere, and then the centers of the eight cubical cells of the hypercube must be projected onto the hypersphere. The 16 vertices of the hypercube plus the projections of the centers of the eight cells of the hypercube form a network of 24 points on the hypersphere. These 24 points are the vertices of the 24-tope, the icosatetratope, the fourth of the six regular 4-D hypersolid polytopes. Curiously, this method also produces a regular figure in two-dimensional space, where it results in an octagon, but in three-dimensional space as well as all spaces higher than four the figure that results is not regular. Having decorated four walls of his room, Percy is still not yet finished, for he has two more walls to decorate -- the in wall and the out wall. He will decorate his in wall with a picture of a hecatonicosatope, which I will make a brave but futile effort to describe next time. Until that next time, take care, and don't try to visualize the fourth dimension -- it's not a visual thing. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Art_Nouveau_Midget { ; time=0:03:48.66 SF5 on P200
; Version 2000 Patchlevel 6
reset=2000 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-3.415040480289464/-2.498805152893876/240\
7.815/1/-127.499 params=1/-1.05/-1/1.05/0/1000
float=y maxiter=2500 bailout=25 inside=0 logmap=55
symmetry=none periodicity=10
colors=000szm<6>QZpMVpIRp<3>2Cq<3>e5Lo3Dx25<6>_73W83\
T93<3>FB3<3>Q90OF3<3>J_CIeEHjHGqJFzLFqJFlHFeFFVE<2>F\
99<15>ma7pc7rd7<3>zk7<16>IUGGTGDSH<3>3OI<10>Z_Wa`Xda\
Z<3>oeb<3>hi_fj_dkZclZalY_mYYlX<2>TcV<3>McTLcTJcSHcS\
GcREcRCeQBfQ<2>CkLDmJDmm<3>EbmF`mFYmFWmGTn<2>GMwHNz<\
15>MPzMQzMQz<2>NQzNQzPPz<12>nIzpIzrHztHzvGzxGzyHz<4>\
zJzzJzzJzzKzzKz<5>zPzzQzzRzzSzzTz<37>zUz
}
frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}
END 20.0 PAR-FORMULA FILE==================================
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